Is there any neighborhood in [0,1] not containing infinitely manypoints in the range of f? No, the points in the range of f are densein [0,1].

Then this simple construct of a function modeled by a family offunctions, "ranges" from zero to one, with that over the domain the"linear" function sees values in the range between and including zeroand one and corresponding to the natural order of the domain, that"ranges" doesn't need quotes, f ranges from zero to one.

Then, those are some simple properties of this construct. Then, thefree mathematical mind simply enough considers conditions as of otheranalytical properties of this function. With n e R, or x e R+ forf_d(x), Int_0->d f_d(x) dx = d/2. This is simply enough the areaunder the line connecting the origin and (d,1). How about for n e R?It, for r e ran(f_d(n)), f_d(rd), looks like f(x) = x, from zero toone, the simple ray of a line with unit slope, from zero to one. Thearea under that is 1/2. But, the "area" under f_oo(n) = 1. Considerlim_d->oo Int_0->d 1/d dx, that equals one. This looks like a flatline infinitesimally greater than zero, the area under which sums toone. So, there are some _interesting_ properties, of f_d(n) = n/d.